Classification of LCD and self-dual codes over a finite non-unital local ring

2025/01/06 by Anup Kushwaha, Indibar Debnath, Kushwaha, Anup +4 · 1 citation
Computer Science · #16L30 #94B05 #Coding theory and cryptography #FOS: Computer and information sciences #Information Theory (cs.IT)

paper · pdf · doi:10.48550/arxiv.2501.03016

openalex publication_date 2025/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This work explores LCD and self-dual codes over a noncommutative non-unital ring E_p= ⟨ r,s ~|~ pr =ps=0,~ r²=r,~ s²=s,~ rs=r,~ sr=s ⟩ of order p² where p is a prime. Initially, we study the monomial equivalence of two free E_p-linear codes. In addition, a necessary and sufficient condition is derived for a free E_p-linear code to be MDS and almost MDS (shortly AMDS). Then, we use these results to classify MDS and AMDS LCD codes over E₂ and E₃ under monomial equivalence for lengths up to 6. Subsequently, we study left self-dual codes over the ring E_p and classify MDS and AMDS left self-dual codes over E₂ and E₃ for lengths up to 12. Finally, we study self-dual codes over the ring E_p and classify MDS and AMDS self-dual codes over E₂ and E₃ for short lengths.

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